In this dissertation, we consider an applied problem, namely, pursuit-evasion games. These
problems are related to robotics, control theory and computer simulations. We want to find
the solution curves of differential equations for pursuit-evasion games, and investigate the
properties of solution curves. First, we define CAT(0) and CAT(K) spaces, and explain
why they are suitable playing fields, that vastly generalize the usual playing field in the
pursuit-evasion literature. Then we prove our existence and uniqueness theorems for continuous
pursuit curves in CAT(K) spaces, as well as our convergence estimates and regularity
theorem.
Pursuit curves are downward gradient curves for the distance from a moving evader, that
is, for a time-dependent gradient flow. We consider not only pursuit curves, but also more
general time-dependent gradient flow.